I'm definitely in favour of the side-by-side boxplots for this.
Principally because the horizontal-scale variable "round" is discrete (ordered categorical if you wish). The takeaway is that there is a downward trend in mean/median, but an awful lot of variability (that decreases slightly as round # increases). This is exactly what a boxplot (or series of boxplots) shows, and it does so without clutter.
The top plot has introduced a spurious horizontal scale (unless there is meaning to left side vs. right side of the Round 1 box in the first plot, which is not mentioned). The second plot has those same issues (what is the horizontal scale? Is it pick # overall?).
If the statistical issue in question is "do players picked in an earlier round of the draft tend to have a higher value?", boxplots will answer that; your plots appear to go after the related but different question of "is player value related to draft pick number", with the round # of the draft being incidental to that. For that, a lowess curve on plot #1 would do the job for me.

This post is long over-due. I have been meaning to write about this blog for a long time but never got around to it. It's like the email response you postponed because you want to think before you fire it off. But I received two mentions of it within the last few days, which reminded me I have t...